(*
 * Copyright Data61, CSIRO (ABN 41 687 119 230)
 *
 * SPDX-License-Identifier: BSD-2-Clause
 *)

section ‹Arithmetic lemmas›

theory More_Arithmetic
  imports Main "HOL-Library.Type_Length" "HOL-Library.Bit_Operations"
begin

declare iszero_0 [intro]

declare min.absorb1 [simp] min.absorb2 [simp]

lemma n_less_equal_power_2 [simp]:
  "n < 2 ^ n"
  by (fact less_exp)

lemma min_pm [simp]: "min a b + (a - b) = a"
  for a b :: nat
  by arith

lemma min_pm1 [simp]: "a - b + min a b = a"
  for a b :: nat
  by arith

lemma rev_min_pm [simp]: "min b a + (a - b) = a"
  for a b :: nat
  by arith

lemma rev_min_pm1 [simp]: "a - b + min b a = a"
  for a b :: nat
  by arith

lemma min_minus [simp]: "min m (m - k) = m - k"
  for m k :: nat
  by arith

lemma min_minus' [simp]: "min (m - k) m = m - k"
  for m k :: nat
  by arith

lemma nat_less_power_trans:
  fixes n :: nat
  assumes nv: "n < 2 ^ (m - k)"
  and     kv: "k ≤ m"
  shows "2 ^ k * n < 2 ^ m"
proof (rule order_less_le_trans)
  show "2 ^ k * n < 2 ^ k * 2 ^ (m - k)"
    by (rule mult_less_mono2 [OF nv zero_less_power]) simp
  show "(2::nat) ^ k * 2 ^ (m - k) ≤ 2 ^ m" using nv kv
    by (subst power_add [symmetric]) simp
qed

lemma nat_le_power_trans:
  fixes n :: nat
  shows "⟦n ≤ 2 ^ (m - k); k ≤ m⟧ ⟹ 2 ^ k * n ≤ 2 ^ m"
  by (metis le_add_diff_inverse mult_le_mono2 semiring_normalization_rules(26))

lemma nat_add_offset_less:
  fixes x :: nat
  assumes yv: "y < 2 ^ n"
  and     xv: "x < 2 ^ m"
  and     mn: "sz = m + n"
  shows   "x * 2 ^ n + y < 2 ^ sz"
proof (subst mn)
  from yv obtain qy where "y + qy = 2 ^ n" and "0 < qy"
    by (auto dest: less_imp_add_positive)

  have "x * 2 ^ n + y < x * 2 ^ n + 2 ^ n" by simp fact+
  also have "… = (x + 1) * 2 ^ n" by simp
  also have "… ≤ 2 ^ (m + n)" using xv
    by (subst power_add) (rule mult_le_mono1, simp)
  finally show "x * 2 ^ n + y < 2 ^ (m + n)" .
qed

lemma nat_power_less_diff:
  assumes lt: "(2::nat) ^ n * q < 2 ^ m"
  shows "q < 2 ^ (m - n)"
  using lt
proof (induct n arbitrary: m)
  case 0
  then show ?case by simp
next
  case (Suc n)

  have ih: "⋀m. 2 ^ n * q < 2 ^ m ⟹ q < 2 ^ (m - n)"
    and prem: "2 ^ Suc n * q < 2 ^ m" by fact+

  show ?case
  proof (cases m)
    case 0
    then show ?thesis using Suc by simp
  next
    case (Suc m')
    then show ?thesis using prem
      by (simp add: ac_simps ih)
  qed
qed

lemma power_2_mult_step_le:
  "⟦n' ≤ n; 2 ^ n' * k' < 2 ^ n * k⟧ ⟹ 2 ^ n' * (k' + 1) ≤ 2 ^ n * (k::nat)"
  apply (cases "n'=n", simp)
   apply (metis Suc_leI le_refl mult_Suc_right mult_le_mono semiring_normalization_rules(7))
  apply (drule (1) le_neq_trans)
  apply clarsimp
  apply (subgoal_tac "∃m. n = n' + m")
   prefer 2
   apply (simp add: le_Suc_ex)
  apply (clarsimp simp: power_add)
  apply (metis Suc_leI mult.assoc mult_Suc_right nat_mult_le_cancel_disj)
  done

lemma nat_mult_power_less_eq:
  "b > 0 ⟹ (a * b ^ n < (b :: nat) ^ m) = (a < b ^ (m - n))"
  using mult_less_cancel2[where m = a and k = "b ^ n" and n="b ^ (m - n)"]
        mult_less_cancel2[where m="a * b ^ (n - m)" and k="b ^ m" and n=1]
  apply (simp only: power_add[symmetric] nat_minus_add_max)
  apply (simp only: power_add[symmetric] nat_minus_add_max ac_simps)
  apply (simp add: max_def split: if_split_asm)
  done

lemma diff_diff_less:
  "(i < m - (m - (n :: nat))) = (i < m ∧ i < n)"
  by auto

lemma small_powers_of_2:
  ‹x < 2 ^ (x - 1)› if ‹x ≥ 3› for x :: nat
proof -
  define m where ‹m = x - 3›
  with that have ‹x = m + 3›
    by simp
  moreover have ‹m + 3 < 4 * 2 ^ m›
    by (induction m) simp_all
  ultimately show ?thesis
    by simp
qed

end

(*
 * Copyright Data61, CSIRO (ABN 41 687 119 230)
 *
 * SPDX-License-Identifier: BSD-2-Clause
 *)

section ‹Lemmas on division›

theory More_Divides
  imports
    "HOL-Library.Word"
begin

declare div_eq_dividend_iff [simp]

lemma int_div_same_is_1 [simp]:
  ‹a div b = a ⟷ b = 1› if ‹0 < a› for a b :: int
  using that by (metis div_by_1 abs_ge_zero abs_of_pos int_div_less_self neq_iff
            nonneg1_imp_zdiv_pos_iff zabs_less_one_iff)

lemma int_div_minus_is_minus1 [simp]:
  ‹a div b = - a ⟷ b = - 1› if ‹0 > a› for a b :: int
  using that by (metis div_minus_right equation_minus_iff int_div_same_is_1 neg_0_less_iff_less)

lemma nat_div_eq_Suc_0_iff: "n div m = Suc 0 ⟷ m ≤ n ∧ n < 2 * m"
  apply auto
  using div_greater_zero_iff apply fastforce
   apply (metis One_nat_def div_greater_zero_iff dividend_less_div_times mult.right_neutral mult_Suc mult_numeral_1 numeral_2_eq_2 zero_less_numeral)
  apply (simp add: div_nat_eqI)
  done

lemma diff_mod_le:
  ‹a - a mod b ≤ d - b› if ‹a < d› ‹b dvd d› for a b d :: nat
  using that
  apply(subst minus_mod_eq_mult_div)
  apply(clarsimp simp: dvd_def)
  apply(cases ‹b = 0›)
   apply simp
  apply(subgoal_tac "a div b ≤ k - 1")
   prefer 2
   apply(subgoal_tac "a div b < k")
    apply(simp add: less_Suc_eq_le [symmetric])
   apply(subgoal_tac "b * (a div b) < b * ((b * k) div b)")
    apply clarsimp
   apply(subst div_mult_self1_is_m)
    apply arith
   apply(rule le_less_trans)
    apply simp
    apply(subst mult.commute)
    apply(rule div_times_less_eq_dividend)
   apply assumption
  apply clarsimp
  apply(subgoal_tac "b * (a div b) ≤ b * (k - 1)")
   apply(erule le_trans)
   apply(simp add: diff_mult_distrib2)
  apply simp
  done

lemma one_mod_exp_eq_one [simp]:
  "1 mod (2 * 2 ^ n) = (1::int)"
  using power_gt1 [of 2 n] by (auto intro: mod_pos_pos_trivial)

lemma int_mod_lem: "0 < n ⟹ 0 ≤ b ∧ b < n ⟷ b mod n = b"
  for b n :: int
  apply safe
    apply (erule (1) mod_pos_pos_trivial)
   apply (erule_tac [!] subst)
   apply auto
  done

lemma int_mod_ge': "b < 0 ⟹ 0 < n ⟹ b + n ≤ b mod n"
  for b n :: int
  by (metis add_less_same_cancel2 int_mod_ge mod_add_self2)

lemma int_mod_le': "0 ≤ b - n ⟹ b mod n ≤ b - n"
  for b n :: int
  by (metis minus_mod_self2 zmod_le_nonneg_dividend)

lemma emep1: "even n ⟹ even d ⟹ 0 ≤ d ⟹ (n + 1) mod d = (n mod d) + 1"
  for n d :: int
  by (auto simp add: pos_zmod_mult_2 add.commute dvd_def)

lemma m1mod2k: "- 1 mod 2 ^ n = (2 ^ n - 1 :: int)"
  by (rule zmod_minus1) simp

lemma sb_inc_lem: "a + 2^k < 0 ⟹ a + 2^k + 2^(Suc k) ≤ (a + 2^k) mod 2^(Suc k)"
  for a :: int
  using int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"]
  by simp

lemma sb_inc_lem': "a < - (2^k) ⟹ a + 2^k + 2^(Suc k) ≤ (a + 2^k) mod 2^(Suc k)"
  for a :: int
  by (rule sb_inc_lem) simp

lemma sb_dec_lem: "0 ≤ - (2 ^ k) + a ⟹ (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a"
  for a :: int
  using int_mod_le'[where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"] by simp

lemma sb_dec_lem': "2 ^ k ≤ a ⟹ (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a"
  for a :: int
  by (rule sb_dec_lem) simp

lemma mod_2_neq_1_eq_eq_0: "k mod 2 ≠ 1 ⟷ k mod 2 = 0"
  for k :: int
  by (fact not_mod_2_eq_1_eq_0)

lemma z1pmod2: "(2 * b + 1) mod 2 = (1::int)"
  for b :: int
  by arith

lemma p1mod22k': "(1 + 2 * b) mod (2 * 2 ^ n) = 1 + 2 * (b mod 2 ^ n)"
  for b :: int
  by (rule pos_zmod_mult_2) simp

lemma p1mod22k: "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + 1"
  for b :: int
  by (simp add: p1mod22k' add.commute)

lemma pos_mod_sign2:
  ‹0 ≤ a mod 2› for a :: int
  by simp

lemma pos_mod_bound2:
  ‹a mod 2 < 2› for a :: int
  by simp

lemma nmod2: "n mod 2 = 0 ∨ n mod 2 = 1"
  for n :: int
  by arith

lemma eme1p:
  "even n ⟹ even d ⟹ 0 ≤ d ⟹ (1 + n) mod d = 1 + n mod d" for n d :: int
  using emep1 [of n d] by (simp add: ac_simps)

lemma m1mod22k:
  ‹- 1 mod (2 * 2 ^ n) = 2 * 2 ^ n - (1::int)›
  by (simp add: zmod_minus1)

lemma z1pdiv2: "(2 * b + 1) div 2 = b"
  for b :: int
  by arith

lemma zdiv_le_dividend:
  ‹0 ≤ a ⟹ 0 < b ⟹ a div b ≤ a› for a b :: int
  by (metis div_by_1 int_one_le_iff_zero_less zdiv_mono2 zero_less_one)

lemma axxmod2: "(1 + x + x) mod 2 = 1 ∧ (0 + x + x) mod 2 = 0"
  for x :: int
  by arith

lemma axxdiv2: "(1 + x + x) div 2 = x ∧ (0 + x + x) div 2 = x"
  for x :: int
  by arith

lemmas rdmods =
  mod_minus_eq [symmetric]
  mod_diff_left_eq [symmetric]
  mod_diff_right_eq [symmetric]
  mod_add_left_eq [symmetric]
  mod_add_right_eq [symmetric]
  mod_mult_right_eq [symmetric]
  mod_mult_left_eq [symmetric]

lemma mod_plus_right: "(a + x) mod m = (b + x) mod m ⟷ a mod m = b mod m"
  for a b m x :: nat
  by (induct x) (simp_all add: mod_Suc, arith)

lemma nat_minus_mod: "(n - n mod m) mod m = 0"
  for m n :: nat
  by (induct n) (simp_all add: mod_Suc)

lemmas nat_minus_mod_plus_right =
  trans [OF nat_minus_mod mod_0 [symmetric],
    THEN mod_plus_right [THEN iffD2], simplified]

lemmas push_mods' = mod_add_eq
  mod_mult_eq mod_diff_eq
  mod_minus_eq

lemmas push_mods = push_mods' [THEN eq_reflection]
lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection]

lemma nat_mod_eq: "b < n ⟹ a mod n = b mod n ⟹ a mod n = b"
  for a b n :: nat
  by (induct a) auto

lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq]

lemma nat_mod_lem: "0 < n ⟹ b < n ⟷ b mod n = b"
  for b n :: nat
  apply safe
   apply (erule nat_mod_eq')
  apply (erule subst)
  apply (erule mod_less_divisor)
  done

lemma mod_nat_add: "x < z ⟹ y < z ⟹ (x + y) mod z = (if x + y < z then x + y else x + y - z)"
  for x y z :: nat
  apply (rule nat_mod_eq)
   apply auto
  apply (rule trans)
   apply (rule le_mod_geq)
   apply simp
  apply (rule nat_mod_eq')
  apply arith
  done

lemma mod_nat_sub: "x < z ⟹ (x - y) mod z = x - y"
  for x y :: nat
  by (rule nat_mod_eq') arith

lemma int_mod_eq: "0 ≤ b ⟹ b < n ⟹ a mod n = b mod n ⟹ a mod n = b"
  for a b n :: int
  by (metis mod_pos_pos_trivial)

lemma zmde:
  ‹b * (a div b) = a - a mod b› for a b :: ‹'a::{group_add,semiring_modulo}›
  using mult_div_mod_eq [of b a] by (simp add: eq_diff_eq)

(* already have this for naturals, div_mult_self1/2, but not for ints *)
lemma zdiv_mult_self: "m ≠ 0 ⟹ (a + m * n) div m = a div m + n"
  for a m n :: int
  by simp

lemma mod_power_lem: "a > 1 ⟹ a ^ n mod a ^ m = (if m ≤ n then 0 else a ^ n)"
  for a :: int
  by (simp add: mod_eq_0_iff_dvd le_imp_power_dvd)

lemma nonneg_mod_div: "0 ≤ a ⟹ 0 ≤ b ⟹ 0 ≤ (a mod b) ∧ 0 ≤ a div b"
  for a b :: int
  by (cases "b = 0") (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2])

lemma mod_exp_less_eq_exp:
  ‹a mod 2 ^ n < 2 ^ n› for a :: int
  by (rule pos_mod_bound) simp

lemma div_mult_le:
  ‹a div b * b ≤ a› for a b :: nat
  by (fact div_times_less_eq_dividend)

lemma power_sub:
  fixes a :: nat
  assumes lt: "n ≤ m"
  and     av: "0 < a"
  shows "a ^ (m - n) = a ^ m div a ^ n"
proof (subst nat_mult_eq_cancel1 [symmetric])
  show "(0::nat) < a ^ n" using av by simp
next
  from lt obtain q where mv: "n + q = m"
    by (auto simp: le_iff_add)

  have "a ^ n * (a ^ m div a ^ n) = a ^ m"
  proof (subst mult.commute)
    have "a ^ m = (a ^ m div a ^ n) * a ^ n + a ^ m mod a ^ n"
      by (rule  div_mult_mod_eq [symmetric])

    moreover have "a ^ m mod a ^ n = 0"
      by (subst mod_eq_0_iff_dvd, subst dvd_def, rule exI [where x = "a ^ q"],
      (subst power_add [symmetric] mv)+, rule refl)

    ultimately show "(a ^ m div a ^ n) * a ^ n = a ^ m" by simp
  qed

  then show "a ^ n * a ^ (m - n) = a ^ n * (a ^ m div a ^ n)" using lt
    by (simp add: power_add [symmetric])
qed

lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
  apply (cut_tac m = q and n = c in mod_less_divisor)
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
  apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)
  apply (simp add: add_mult_distrib2)
  done

lemma less_two_pow_divD:
  "⟦ (x :: nat) < 2 ^ n div 2 ^ m ⟧
    ⟹ n ≥ m ∧ (x < 2 ^ (n - m))"
  apply (rule context_conjI)
   apply (rule ccontr)
   apply (simp add: power_strict_increasing)
  apply (simp add: power_sub)
  done

lemma less_two_pow_divI:
  "⟦ (x :: nat) < 2 ^ (n - m); m ≤ n ⟧ ⟹ x < 2 ^ n div 2 ^ m"
  by (simp add: power_sub)

lemmas m2pths = pos_mod_sign mod_exp_less_eq_exp

lemmas int_mod_eq' = mod_pos_pos_trivial (* FIXME delete *)

lemmas int_mod_le = zmod_le_nonneg_dividend (* FIXME: delete *)

lemma power_mod_div:
  fixes x :: "nat"
  shows "x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)" (is "?LHS = ?RHS")
proof (cases "n ≤ m")
  case True
  then have "?LHS = 0"
    apply -
    apply (rule div_less)
    apply (rule order_less_le_trans [OF mod_less_divisor]; simp)
    done
  also have "… = ?RHS" using True
    by simp
  finally show ?thesis .
next
  case False
  then have lt: "m < n" by simp
  then obtain q where nv: "n = m + q" and "0 < q"
    by (auto dest: less_imp_Suc_add)

  then have "x mod 2 ^ n = 2 ^ m * (x div 2 ^ m mod 2 ^ q) + x mod 2 ^ m"
    by (simp add: power_add mod_mult2_eq)

  then have "?LHS = x div 2 ^ m mod 2 ^ q"
    by (simp add: div_add1_eq)

  also have "… = ?RHS" using nv
    by simp

  finally show ?thesis .
qed

lemma mod_mod_power:
  fixes k :: nat
  shows "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ (min m n)"
proof (cases "m ≤ n")
  case True

  then have "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ m"
    apply -
    apply (subst mod_less [where n = "2 ^ n"])
    apply (rule order_less_le_trans [OF mod_less_divisor])
    apply simp+
    done
  also have "… = k mod  2 ^ (min m n)" using True by simp
  finally show ?thesis .
next
  case False
  then have "n < m" by simp
  then obtain d where md: "m = n + d"
    by (auto dest: less_imp_add_positive)
  then have "k mod 2 ^ m = 2 ^ n * (k div 2 ^ n mod 2 ^ d) + k mod 2 ^ n"
    by (simp add: mod_mult2_eq power_add)
  then have "k mod 2 ^ m mod 2 ^ n = k mod 2 ^ n"
    by (simp add: mod_add_left_eq)
  then show ?thesis using False
    by simp
qed

lemma mod_div_equality_div_eq:
  "a div b * b = (a - (a mod b) :: int)"
  by (simp add: field_simps)

lemma zmod_helper:
  "n mod m = k ⟹ ((n :: int) + a) mod m = (k + a) mod m"
  by (metis add.commute mod_add_right_eq)

lemma int_div_sub_1:
  "⟦ m ≥ 1 ⟧ ⟹ (n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)"
  apply (subgoal_tac "m = 0 ∨ (n - (1 :: int)) div m = (if m dvd n then (n div m) - 1 else n div m)")
   apply fastforce
  apply (subst mult_cancel_right[symmetric])
  apply (simp only: left_diff_distrib split: if_split)
  apply (simp only: mod_div_equality_div_eq)
  apply (clarsimp simp: field_simps)
  apply (clarsimp simp: dvd_eq_mod_eq_0)
  apply (cases "m = 1")
   apply simp
  apply (subst mod_diff_eq[symmetric], simp add: zmod_minus1)
  apply clarsimp
  apply (subst diff_add_cancel[where b=1, symmetric])
  apply (subst mod_add_eq[symmetric])
  apply (simp add: field_simps)
  apply (rule mod_pos_pos_trivial)
   apply (subst add_0_right[where a=0, symmetric])
   apply (rule add_mono)
    apply simp
   apply simp
  apply (cases "(n - 1) mod m = m - 1")
   apply (drule zmod_helper[where a=1])
   apply simp
  apply (subgoal_tac "1 + (n - 1) mod m ≤ m")
   apply simp
  apply (subst field_simps, rule zless_imp_add1_zle)
  apply simp
  done

lemma power_minus_is_div:
  "b ≤ a ⟹ (2 :: nat) ^ (a - b) = 2 ^ a div 2 ^ b"
  apply (induct a arbitrary: b)
   apply simp
  apply (erule le_SucE)
   apply (clarsimp simp:Suc_diff_le le_iff_add power_add)
  apply simp
  done

lemma two_pow_div_gt_le:
  "v < 2 ^ n div (2 ^ m :: nat) ⟹ m ≤ n"
  by (clarsimp dest!: less_two_pow_divD)

lemma td_gal_lt:
  ‹0 < c ⟹ a < b * c ⟷ a div c < b›
  for a b c :: nat
  apply (auto dest: less_mult_imp_div_less)
  apply (metis div_le_mono div_mult_self_is_m leD leI)
  done

lemma td_gal:
  ‹0 < c ⟹ b * c ≤ a  ⟷ b ≤ a div c›
  for a b c :: nat
  by (meson not_le td_gal_lt)

end

(*
 * Copyright Data61, CSIRO (ABN 41 687 119 230)
 *
 * SPDX-License-Identifier: BSD-2-Clause
 *)

section ‹Lemmas on words›

theory More_Word
  imports
    "HOL-Library.Word"
    More_Arithmetic
    More_Divides
begin

lemma unat_power_lower [simp]:
  "unat ((2::'a::len word) ^ n) = 2 ^ n" if "n < LENGTH('a::len)"
  using that by transfer simp

lemma unat_p2: "n < LENGTH('a :: len) ⟹ unat (2 ^ n :: 'a word) = 2 ^ n"
  by (fact unat_power_lower)

lemma word_div_lt_eq_0:
  "x < y ⟹ x div y = 0" for x :: "'a :: len word"
  by transfer simp

lemma word_div_eq_1_iff: "n div m = 1 ⟷ n ≥ m ∧ unat n < 2 * unat (m :: 'a :: len word)"
  apply (simp only: word_arith_nat_defs word_le_nat_alt word_of_nat_eq_iff flip: nat_div_eq_Suc_0_iff)
  apply (simp flip: unat_div unsigned_take_bit_eq)
  done

lemma shiftl_power:
  "(shiftl1 ^^ x) (y::'a::len word) = 2 ^ x * y"
  apply (induct x)
   apply simp
  apply (simp add: shiftl1_2t)
  done

lemma AND_twice [simp]:
  "(w AND m) AND m = w AND m"
  by (fact and.right_idem)

lemma word_combine_masks:
  "w AND m = z ⟹ w AND m' = z' ⟹ w AND (m OR m') = (z OR z')"
  for w m m' z z' :: ‹'a::len word›
  by (simp add: bit.conj_disj_distrib)

lemma p2_gt_0:
  "(0 < (2 ^ n :: 'a :: len word)) = (n < LENGTH('a))"
  by (simp add : word_gt_0 not_le)

lemma uint_2p_alt:
  ‹n < LENGTH('a::len) ⟹ uint ((2::'a::len word) ^ n) = 2 ^ n›
  using p2_gt_0 [of n, where ?'a = 'a] by (simp add: uint_2p)

lemma p2_eq_0:
  ‹(2::'a::len word) ^ n = 0 ⟷ LENGTH('a::len) ≤ n›
  by (fact exp_eq_zero_iff)

lemma p2len:
  ‹(2 :: 'a word) ^ LENGTH('a::len) = 0›
  by simp

lemma neg_mask_is_div:
  "w AND NOT (mask n) = (w div 2^n) * 2^n"
  for w :: ‹'a::len word›
  by (rule bit_word_eqI)
    (auto simp add: bit_simps simp flip: push_bit_eq_mult drop_bit_eq_div)

lemma neg_mask_is_div':
  "n < size w ⟹ w AND NOT (mask n) = ((w div (2 ^ n)) * (2 ^ n))"
  for w :: ‹'a::len word›
  by (rule neg_mask_is_div)

lemma and_mask_arith:
  "w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
  for w :: ‹'a::len word›
  by (rule bit_word_eqI)
    (auto simp add: bit_simps word_size simp flip: push_bit_eq_mult drop_bit_eq_div)

lemma and_mask_arith':
  "0 < n ⟹ w AND mask n = (w * 2^(size w - n)) div 2^(size w - n)"
  for w :: ‹'a::len word›
  by (rule and_mask_arith)

lemma mask_2pm1: "mask n = 2 ^ n - (1 :: 'a::len word)"
  by (fact mask_eq_decr_exp)

lemma add_mask_fold:
  "x + 2 ^ n - 1 = x + mask n"
  for x :: ‹'a::len word›
  by (simp add: mask_eq_decr_exp)

lemma word_and_mask_le_2pm1: "w AND mask n ≤ 2 ^ n - 1"
  for w :: ‹'a::len word›
  by (simp add: mask_2pm1[symmetric] word_and_le1)

lemma is_aligned_AND_less_0:
  "u AND mask n = 0 ⟹ v < 2^n ⟹ u AND v = 0"
  for u v :: ‹'a::len word›
  apply (drule less_mask_eq)
  apply (simp flip: take_bit_eq_mask)
  apply (simp add: bit_eq_iff)
  apply (auto simp add: bit_simps)
  done

lemma le_shiftr1:
  ‹shiftr1 u ≤ shiftr1 v› if ‹u ≤ v›
using that proof transfer
  fix k l :: int
  assume ‹take_bit LENGTH('a) k ≤ take_bit LENGTH('a) l›
  then have ‹take_bit LENGTH('a) (drop_bit 1 (take_bit LENGTH('a) k))
    ≤ take_bit LENGTH('a) (drop_bit 1 (take_bit LENGTH('a) l))›
    apply (simp add: take_bit_drop_bit min_def)
    apply (simp add: drop_bit_eq_div)
    done
  then show ‹take_bit LENGTH('a) (take_bit LENGTH('a) k div 2)
    ≤ take_bit LENGTH('a) (take_bit LENGTH('a) l div 2)›
    by (simp add: drop_bit_eq_div)
qed

lemma and_mask_eq_iff_le_mask:
  ‹w AND mask n = w ⟷ w ≤ mask n›
  for w :: ‹'a::len word›
  apply (simp flip: take_bit_eq_mask)
  apply (cases ‹n ≥ LENGTH('a)›; transfer)
   apply (simp_all add: not_le min_def)
   apply (simp_all add: mask_eq_exp_minus_1)
  apply auto
   apply (metis take_bit_int_less_exp)
  apply (metis min_def nat_less_le take_bit_int_eq_self_iff take_bit_take_bit)
  done

lemma less_eq_mask_iff_take_bit_eq_self:
  ‹w ≤ mask n ⟷ take_bit n w = w›
  for w :: ‹'a::len word›
  by (simp add: and_mask_eq_iff_le_mask take_bit_eq_mask)

lemma NOT_eq:
  "NOT (x :: 'a :: len word) = - x - 1"
  apply (cut_tac x = "x" in word_add_not)
  apply (drule add.commute [THEN trans])
  apply (drule eq_diff_eq [THEN iffD2])
  by simp

lemma NOT_mask: "NOT (mask n :: 'a::len word) = - (2 ^ n)"
  by (simp add : NOT_eq mask_2pm1)

lemma le_m1_iff_lt: "(x > (0 :: 'a :: len word)) = ((y ≤ x - 1) = (y < x))"
  by uint_arith

lemma gt0_iff_gem1:
  ‹0 < x ⟷ x - 1 < x›
  for x :: ‹'a::len word›
  by (metis add.right_neutral diff_add_cancel less_irrefl measure_unat unat_arith_simps(2) word_neq_0_conv word_sub_less_iff)

lemma power_2_ge_iff:
  ‹2 ^ n - (1 :: 'a::len word) < 2 ^ n ⟷ n < LENGTH('a)›
  using gt0_iff_gem1 p2_gt_0 by blast

lemma le_mask_iff_lt_2n:
  "n < len_of TYPE ('a) = (((w :: 'a :: len word) ≤ mask n) = (w < 2 ^ n))"
  unfolding mask_2pm1 by (rule trans [OF p2_gt_0 [THEN sym] le_m1_iff_lt])

lemma mask_lt_2pn:
  ‹n < LENGTH('a) ⟹ mask n < (2 :: 'a::len word) ^ n›
  by (simp add: mask_eq_exp_minus_1 power_2_ge_iff)

lemma word_unat_power:
  "(2 :: 'a :: len word) ^ n = of_nat (2 ^ n)"
  by simp

lemma of_nat_mono_maybe:
  assumes xlt: "x < 2 ^ len_of TYPE ('a)"
  shows   "y < x ⟹ of_nat y < (of_nat x :: 'a :: len word)"
  apply (subst word_less_nat_alt)
  apply (subst unat_of_nat)+
  apply (subst mod_less)
   apply (erule order_less_trans [OF _ xlt])
  apply (subst mod_less [OF xlt])
  apply assumption
  done

lemma word_and_max_word:
  fixes a::"'a::len word"
  shows "x = max_word ⟹ a AND x = a"
  by simp

lemma word_and_full_mask_simp:
  ‹x AND mask LENGTH('a) = x› for x :: ‹'a::len word›
proof (rule bit_eqI)
  fix n
  assume ‹2 ^ n ≠ (0 :: 'a word)›
  then have ‹n < LENGTH('a)›
    by simp
  then show ‹bit (x AND Bit_Operations.mask LENGTH('a)) n ⟷ bit x n›
    by (simp add: bit_and_iff bit_mask_iff)
qed

lemma of_int_uint:
  "of_int (uint x) = x"
  by (fact word_of_int_uint)

corollary word_plus_and_or_coroll:
  "x AND y = 0 ⟹ x + y = x OR y"
  for x y :: ‹'a::len word›
  using word_plus_and_or[where x=x and y=y]
  by simp

corollary word_plus_and_or_coroll2:
  "(x AND w) + (x AND NOT w) = x"
  for x w :: ‹'a::len word›
  apply (subst disjunctive_add)
   apply (simp add: bit_simps)
  apply (simp flip: bit.conj_disj_distrib)
  done

lemma nat_mask_eq:
  ‹nat (mask n) = mask n›
  by (simp add: nat_eq_iff of_nat_mask_eq)

lemma unat_mask_eq:
  ‹unat (mask n :: 'a::len word) = mask (min LENGTH('a) n)›
  by transfer (simp add: nat_mask_eq)

lemma word_plus_mono_left:
  fixes x :: "'a :: len word"
  shows "⟦y ≤ z; x ≤ x + z⟧ ⟹ y + x ≤ z + x"
  by unat_arith

lemma less_Suc_unat_less_bound:
  "n < Suc (unat (x :: 'a :: len word)) ⟹ n < 2 ^ LENGTH('a)"
  by (auto elim!: order_less_le_trans intro: Suc_leI)

lemma up_ucast_inj:
  "⟦ ucast x = (ucast y::'b::len word); LENGTH('a) ≤ len_of TYPE ('b) ⟧ ⟹ x = (y::'a::len word)"
  by transfer (simp add: min_def split: if_splits)

lemmas ucast_up_inj = up_ucast_inj

lemma up_ucast_inj_eq:
  "LENGTH('a) ≤ len_of TYPE ('b) ⟹ (ucast x = (ucast y::'b::len word)) = (x = (y::'a::len word))"
  by (fastforce dest: up_ucast_inj)

lemma no_plus_overflow_neg:
  "(x :: 'a :: len word) < -y ⟹ x ≤ x + y"
  by (metis diff_minus_eq_add less_imp_le sub_wrap_lt)

lemma ucast_ucast_eq:
  "⟦ ucast x = (ucast (ucast y::'a word)::'c::len word); LENGTH('a) ≤ LENGTH('b);
     LENGTH('b) ≤ LENGTH('c) ⟧ ⟹
   x = ucast y" for x :: "'a::len word" and y :: "'b::len word"
  apply transfer
  apply (cases ‹LENGTH('c) = LENGTH('a)›)
   apply (auto simp add: min_def)
  done

lemma ucast_0_I:
  "x = 0 ⟹ ucast x = 0"
  by simp

lemma word_add_offset_less:
  fixes x :: "'a :: len word"
  assumes yv: "y < 2 ^ n"
  and     xv: "x < 2 ^ m"
  and     mnv: "sz < LENGTH('a :: len)"
  and    xv': "x < 2 ^ (LENGTH('a :: len) - n)"
  and     mn: "sz = m + n"
  shows   "x * 2 ^ n + y < 2 ^ sz"
proof (subst mn)
  from mnv mn have nv: "n < LENGTH('a)" and mv: "m < LENGTH('a)"  by auto

  have uy: "unat y < 2 ^ n"
    by (rule order_less_le_trans [OF unat_mono [OF yv] order_eq_refl],
        rule unat_power_lower[OF nv])

  have ux: "unat x < 2 ^ m"
    by (rule order_less_le_trans [OF unat_mono [OF xv] order_eq_refl],
        rule unat_power_lower[OF mv])

  then show "x * 2 ^ n + y < 2 ^ (m + n)" using ux uy nv mnv xv'
    apply (subst word_less_nat_alt)
    apply (subst unat_word_ariths)+
    apply (subst mod_less)
     apply simp
     apply (subst mult.commute)
     apply (rule nat_less_power_trans [OF _ order_less_imp_le [OF nv]])
     apply (rule order_less_le_trans [OF unat_mono [OF xv']])
     apply (cases "n = 0"; simp)
    apply (subst unat_power_lower[OF nv])
    apply (subst mod_less)
     apply (erule order_less_le_trans [OF nat_add_offset_less], assumption)
      apply (rule mn)
     apply simp
    apply (simp add: mn mnv)
    apply (erule nat_add_offset_less; simp)
    done
qed

lemma word_less_power_trans:
  fixes n :: "'a :: len word"
  assumes nv: "n < 2 ^ (m - k)"
  and     kv: "k ≤ m"
  and     mv: "m < len_of TYPE ('a)"
  shows "2 ^ k * n < 2 ^ m"
  using nv kv mv
  apply -
  apply (subst word_less_nat_alt)
  apply (subst unat_word_ariths)
  apply (subst mod_less)
   apply simp
   apply (rule nat_less_power_trans)
    apply (erule order_less_trans [OF unat_mono])
    apply simp
   apply simp
  apply simp
  apply (rule nat_less_power_trans)
   apply (subst unat_power_lower[where 'a = 'a, symmetric])
    apply simp
   apply (erule unat_mono)
  apply simp
  done

lemma  word_less_power_trans2:
  fixes n :: "'a::len word"
  shows "⟦n < 2 ^ (m - k); k ≤ m; m < LENGTH('a)⟧ ⟹ n * 2 ^ k < 2 ^ m"
  by (subst field_simps, rule word_less_power_trans)

lemma Suc_unat_diff_1:
  fixes x :: "'a :: len word"
  assumes lt: "1 ≤ x"
  shows "Suc (unat (x - 1)) = unat x"
proof -
  have "0 < unat x"
    by (rule order_less_le_trans [where y = 1], simp, subst unat_1 [symmetric],
        rule iffD1 [OF word_le_nat_alt lt])

  then show ?thesis
    by ((subst unat_sub [OF lt])+, simp only:  unat_1)
qed

lemma word_eq_unatI:
  ‹v = w› if ‹unat v = unat w›
  using that by transfer (simp add: nat_eq_iff)

lemma word_div_sub:
  fixes x :: "'a :: len word"
  assumes yx: "y ≤ x"
  and     y0: "0 < y"
  shows "(x - y) div y = x div y - 1"
  apply (rule word_eq_unatI)
  apply (subst unat_div)
  apply (subst unat_sub [OF yx])
  apply (subst unat_sub)
   apply (subst word_le_nat_alt)
   apply (subst unat_div)
   apply (subst le_div_geq)
     apply (rule order_le_less_trans [OF _ unat_mono [OF y0]])
     apply simp
    apply (subst word_le_nat_alt [symmetric], rule yx)
   apply simp
  apply (subst unat_div)
  apply (subst le_div_geq [OF _ iffD1 [OF word_le_nat_alt yx]])
   apply (rule order_le_less_trans [OF _ unat_mono [OF y0]])
   apply simp
  apply simp
  done

lemma word_mult_less_mono1:
  fixes i :: "'a :: len word"
  assumes ij: "i < j"
  and    knz: "0 < k"
  and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
  shows  "i * k < j * k"
proof -
  from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)"
    by (auto intro: order_less_subst2 simp: word_less_nat_alt elim: mult_less_mono1)

  then show ?thesis using ujk knz ij
    by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem])
qed

lemma word_mult_less_dest:
  fixes i :: "'a :: len word"
  assumes ij: "i * k < j * k"
  and    uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
  and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
  shows  "i < j"
  using uik ujk ij
  by (auto simp: word_less_nat_alt iffD1 [OF unat_mult_lem] elim: mult_less_mono1)

lemma word_mult_less_cancel:
  fixes k :: "'a :: len word"
  assumes knz: "0 < k"
  and    uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
  and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
  shows "(i * k < j * k) = (i < j)"
  by (rule iffI [OF word_mult_less_dest [OF _ uik ujk] word_mult_less_mono1 [OF _ knz ujk]])

lemma Suc_div_unat_helper:
  assumes szv: "sz < LENGTH('a :: len)"
  and   usszv: "us ≤ sz"
  shows "2 ^ (sz - us) = Suc (unat (((2::'a :: len word) ^ sz - 1) div 2 ^ us))"
proof -
  note usv = order_le_less_trans [OF usszv szv]

  from usszv obtain q where qv: "sz = us + q" by (auto simp: le_iff_add)

  have "Suc (unat (((2:: 'a word) ^ sz - 1) div 2 ^ us)) =
    (2 ^ us + unat ((2:: 'a word) ^ sz - 1)) div 2 ^ us"
    apply (subst unat_div unat_power_lower[OF usv])+
    apply (subst div_add_self1, simp+)
    done

  also have "… = ((2 ^ us - 1) + 2 ^ sz) div 2 ^ us" using szv
    by (simp add: unat_minus_one)

  also have "… = 2 ^ q + ((2 ^ us - 1) div 2 ^ us)"
    apply (subst qv)
    apply (subst power_add)
    apply (subst div_mult_self2; simp)
    done

  also have "… = 2 ^ (sz - us)" using qv by simp

  finally show ?thesis ..
qed

lemma enum_word_nth_eq:
  ‹(Enum.enum :: 'a::len word list) ! n = word_of_nat n›
    if ‹n < 2 ^ LENGTH('a)›
    for n
  using that by (simp add: enum_word_def)

lemma length_enum_word_eq:
  ‹length (Enum.enum :: 'a::len word list) = 2 ^ LENGTH('a)›
  by (simp add: enum_word_def)

lemma unat_lt2p [iff]:
  ‹unat x < 2 ^ LENGTH('a)› for x :: ‹'a::len word›
  by transfer simp

lemma of_nat_unat [simp]:
  "of_nat ∘ unat = id"
  by (rule ext, simp)

lemma Suc_unat_minus_one [simp]:
  "x ≠ 0 ⟹ Suc (unat (x - 1)) = unat x"
  by (metis Suc_diff_1 unat_gt_0 unat_minus_one)

lemma word_add_le_dest:
  fixes i :: "'a :: len word"
  assumes le: "i + k ≤ j + k"
  and    uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
  and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
  shows  "i ≤ j"
  using uik ujk le
  by (auto simp: word_le_nat_alt iffD1 [OF unat_add_lem] elim: add_le_mono1)

lemma word_add_le_mono1:
  fixes i :: "'a :: len word"
  assumes ij: "i ≤ j"
  and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
  shows  "i + k ≤ j + k"
proof -
  from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)"
    by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: add_le_mono1)

  then show ?thesis using ujk ij
    by (auto simp: word_le_nat_alt iffD1 [OF unat_add_lem])
qed

lemma word_add_le_mono2:
  fixes i :: "'a :: len word"
  shows "⟦i ≤ j; unat j + unat k < 2 ^ LENGTH('a)⟧ ⟹ k + i ≤ k + j"
  by (subst field_simps, subst field_simps, erule (1) word_add_le_mono1)

lemma word_add_le_iff:
  fixes i :: "'a :: len word"
  assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
  and     ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
  shows  "(i + k ≤ j + k) = (i ≤ j)"
proof
  assume "i ≤ j"
  show "i + k ≤ j + k" by (rule word_add_le_mono1) fact+
next
  assume "i + k ≤ j + k"
  show "i ≤ j" by (rule word_add_le_dest) fact+
qed

lemma word_add_less_mono1:
  fixes i :: "'a :: len word"
  assumes ij: "i < j"
  and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
  shows  "i + k < j + k"
proof -
  from ij ujk have jk: "unat i + unat k < 2 ^ len_of TYPE ('a)"
    by (auto elim: order_le_less_subst2 simp: word_less_nat_alt elim: add_less_mono1)

  then show ?thesis using ujk ij
    by (auto simp: word_less_nat_alt iffD1 [OF unat_add_lem])
qed

lemma word_add_less_dest:
  fixes i :: "'a :: len word"
  assumes le: "i + k < j + k"
  and    uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
  and    ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
  shows  "i < j"
  using uik ujk le
  by (auto simp: word_less_nat_alt iffD1 [OF unat_add_lem] elim: add_less_mono1)

lemma word_add_less_iff:
  fixes i :: "'a :: len word"
  assumes uik: "unat i + unat k < 2 ^ len_of TYPE ('a)"
  and     ujk: "unat j + unat k < 2 ^ len_of TYPE ('a)"
  shows  "(i + k < j + k) = (i < j)"
proof
  assume "i < j"
  show "i + k < j + k" by (rule word_add_less_mono1) fact+
next
  assume "i + k < j + k"
  show "i < j" by (rule word_add_less_dest) fact+
qed

lemma word_mult_less_iff:
  fixes i :: "'a :: len word"
  assumes knz: "0 < k"
  and     uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
  and     ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
  shows  "(i * k < j * k) = (i < j)"
  using assms by (rule word_mult_less_cancel)

lemma word_le_imp_diff_le:
  fixes n :: "'a::len word"
  shows "⟦k ≤ n; n ≤ m⟧ ⟹ n - k ≤ m"
  by (auto simp: unat_sub word_le_nat_alt)

lemma word_less_imp_diff_less:
  fixes n :: "'a::len word"
  shows "⟦k ≤ n; n < m⟧ ⟹ n - k < m"
  by (clarsimp simp: unat_sub word_less_nat_alt
             intro!: less_imp_diff_less)

lemma word_mult_le_mono1:
  fixes i :: "'a :: len word"
  assumes ij: "i ≤ j"
  and    knz: "0 < k"
  and    ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
  shows  "i * k ≤ j * k"
proof -
  from ij ujk knz have jk: "unat i * unat k < 2 ^ len_of TYPE ('a)"
    by (auto elim: order_le_less_subst2 simp: word_le_nat_alt elim: mult_le_mono1)

  then show ?thesis using ujk knz ij
    by (auto simp: word_le_nat_alt iffD1 [OF unat_mult_lem])
qed

lemma word_mult_le_iff:
  fixes i :: "'a :: len word"
  assumes knz: "0 < k"
  and     uik: "unat i * unat k < 2 ^ len_of TYPE ('a)"
  and     ujk: "unat j * unat k < 2 ^ len_of TYPE ('a)"
  shows  "(i * k ≤ j * k) = (i ≤ j)"
proof
  assume "i ≤ j"
  show "i * k ≤ j * k" by (rule word_mult_le_mono1) fact+
next
  assume p: "i * k ≤ j * k"

  have "0 < unat k" using knz by (simp add: word_less_nat_alt)
  then show "i ≤ j" using p
    by (clarsimp simp: word_le_nat_alt iffD1 [OF unat_mult_lem uik]
      iffD1 [OF unat_mult_lem ujk])
qed

lemma word_diff_less:
  fixes n :: "'a :: len word"
  shows "⟦0 < n; 0 < m; n ≤ m⟧ ⟹ m - n < m"
  apply (subst word_less_nat_alt)
  apply (subst unat_sub)
   apply assumption
  apply (rule diff_less)
   apply (simp_all add: word_less_nat_alt)
  done

lemma word_add_increasing:
  fixes x :: "'a :: len word"
  shows "⟦ p + w ≤ x; p ≤ p + w ⟧ ⟹ p ≤ x"
  by unat_arith

lemma word_random:
  fixes x :: "'a :: len word"
  shows "⟦ p ≤ p + x'; x ≤ x' ⟧ ⟹ p ≤ p + x"
  by unat_arith

lemma word_sub_mono:
  "⟦ a ≤ c; d ≤ b; a - b ≤ a; c - d ≤ c ⟧
    ⟹ (a - b) ≤ (c - d :: 'a :: len word)"
  by unat_arith

lemma power_not_zero:
  "n < LENGTH('a::len) ⟹ (2 :: 'a word) ^ n ≠ 0"
  by (metis p2_gt_0 word_neq_0_conv)

lemma word_gt_a_gt_0:
  "a < n ⟹ (0 :: 'a::len word) < n"
  apply (case_tac "n = 0")
   apply clarsimp
  apply (clarsimp simp: word_neq_0_conv)
  done

lemma word_power_less_1 [simp]:
  "sz < LENGTH('a::len) ⟹ (2::'a word) ^ sz - 1 < 2 ^ sz"
  apply (simp add: word_less_nat_alt)
  apply (subst unat_minus_one)
  apply simp_all
  done

lemma word_sub_1_le:
  "x ≠ 0 ⟹ x - 1 ≤ (x :: ('a :: len) word)"
  apply (subst no_ulen_sub)
  apply simp
  apply (cases "uint x = 0")
   apply (simp add: uint_0_iff)
  apply (insert uint_ge_0[where x=x])
  apply arith
  done

lemma push_bit_word_eq_nonzero:
  ‹push_bit n w ≠ 0› if ‹w < 2 ^ m› ‹m + n < LENGTH('a)› ‹w ≠ 0›
    for w :: ‹'a::len word›
  using that
  apply (simp only: word_neq_0_conv word_less_nat_alt
                    mod_0 unat_word_ariths
                    unat_power_lower word_le_nat_alt)
  apply (metis add_diff_cancel_right' gr0I gr_implies_not0 less_or_eq_imp_le min_def push_bit_eq_0_iff take_bit_nat_eq_self_iff take_bit_push_bit take_bit_take_bit unsigned_push_bit_eq)
  done

lemma unat_less_power:
  fixes k :: "'a::len word"
  assumes szv: "sz < LENGTH('a)"
  and     kv:  "k < 2 ^ sz"
  shows   "unat k < 2 ^ sz"
  using szv unat_mono [OF kv] by simp

lemma unat_mult_power_lem:
  assumes kv: "k < 2 ^ (LENGTH('a::len) - sz)"
  shows "unat (2 ^ sz * of_nat k :: (('a::len) word)) = 2 ^ sz * k"
proof (cases ‹sz < LENGTH('a)›)
  case True
  with assms show ?thesis
    by (simp add: unat_word_ariths take_bit_eq_mod mod_simps)
      (simp add: take_bit_nat_eq_self_iff nat_less_power_trans flip: take_bit_eq_mod)
next
  case False
  with assms show ?thesis
    by simp
qed

lemma word_plus_mcs_4:
  "⟦v + x ≤ w + x; x ≤ v + x⟧ ⟹ v ≤ (w::'a::len word)"
  by uint_arith

lemma word_plus_mcs_3:
  "⟦v ≤ w; x ≤ w + x⟧ ⟹ v + x ≤ w + (x::'a::len word)"
  by unat_arith

lemma word_le_minus_one_leq:
  "x < y ⟹ x ≤ y - 1" for x :: "'a :: len word"
  by transfer (metis le_less_trans less_irrefl take_bit_decr_eq take_bit_nonnegative zle_diff1_eq)

lemma word_less_sub_le[simp]:
  fixes x :: "'a :: len word"
  assumes nv: "n < LENGTH('a)"
  shows "(x ≤ 2 ^ n - 1) = (x < 2 ^ n)"
  using le_less_trans word_le_minus_one_leq nv power_2_ge_iff by blast

lemma unat_of_nat_len:
  "x < 2 ^ LENGTH('a) ⟹ unat (of_nat x :: 'a::len word) = x"
  by (simp add: take_bit_nat_eq_self_iff)

lemma unat_of_nat_eq:
  "x < 2 ^ LENGTH('a) ⟹ unat (of_nat x ::'a::len word) = x"
  by (rule unat_of_nat_len)

lemma unat_eq_of_nat:
  "n < 2 ^ LENGTH('a) ⟹ (unat (x :: 'a::len word) = n) = (x = of_nat n)"
  by transfer
    (auto simp add: take_bit_of_nat nat_eq_iff take_bit_nat_eq_self_iff intro: sym)

lemma alignUp_div_helper:
  fixes a :: "'a::len word"
  assumes kv: "k < 2 ^ (LENGTH('a) - n)"
  and     xk: "x = 2 ^ n * of_nat k"
  and    le: "a ≤ x"
  and    sz: "n < LENGTH('a)"
  and   anz: "a mod 2 ^ n ≠ 0"
  shows "a div 2 ^ n < of_nat k"
proof -
  have kn: "unat (of_nat k :: 'a word) * unat ((2::'a word) ^ n) < 2 ^ LENGTH('a)"
    using xk kv sz
    apply (subst unat_of_nat_eq)
     apply (erule order_less_le_trans)
     apply simp
    apply (subst unat_power_lower, simp)
    apply (subst mult.commute)
    apply (rule nat_less_power_trans)
     apply simp
    apply simp
    done

  have "unat a div 2 ^ n * 2 ^ n ≠ unat a"
  proof -
    have "unat a = unat a div 2 ^ n * 2 ^ n + unat a mod 2 ^ n"
      by (simp add: div_mult_mod_eq)
    also have "… ≠ unat a div 2 ^ n * 2 ^ n" using sz anz
      by (simp add: unat_arith_simps)
    finally show ?thesis ..
  qed

  then have "a div 2 ^ n * 2 ^ n < a" using sz anz
    apply (subst word_less_nat_alt)
    apply (subst unat_word_ariths)
    apply (subst unat_div)
    apply simp
    apply (rule order_le_less_trans [OF mod_less_eq_dividend])
    apply (erule order_le_neq_trans [OF div_mult_le])
    done

  also from xk le have "… ≤ of_nat k * 2 ^ n" by (simp add: field_simps)
  finally show ?thesis using sz kv
    apply -
    apply (erule word_mult_less_dest [OF _ _ kn])
    apply (simp add: unat_div)
    apply (rule order_le_less_trans [OF div_mult_le])
    apply (rule unat_lt2p)
    done
qed

lemma mask_out_sub_mask:
  "(x AND NOT (mask n)) = x - (x AND (mask n))"
  for x :: ‹'a::len word›
  by (simp add: field_simps word_plus_and_or_coroll2)

lemma subtract_mask:
  "p - (p AND mask n) = (p AND NOT (mask n))"
  "p - (p AND NOT (mask n)) = (p AND mask n)"
  for p :: ‹'a::len word›
  by (simp add: field_simps word_plus_and_or_coroll2)+

lemma take_bit_word_eq_self_iff:
  ‹take_bit n w = w ⟷ n ≥ LENGTH('a) ∨ w < 2 ^ n›
  for w :: ‹'a::len word›
  using take_bit_int_eq_self_iff [of n ‹take_bit LENGTH('a) (uint w)›]
  by (transfer fixing: n) auto

lemma word_power_increasing:
  assumes x: "2 ^ x < (2 ^ y::'a::len word)" "x < LENGTH('a::len)" "y < LENGTH('a::len)"
  shows "x < y" using x
  using assms by transfer simp

lemma mask_twice:
  "(x AND mask n) AND mask m = x AND mask (min m n)"
  for x :: ‹'a::len word›
  by (simp flip: take_bit_eq_mask)

lemma plus_one_helper[elim!]:
  "x < n + (1 :: 'a :: len word) ⟹ x ≤ n"
  apply (simp add: word_less_nat_alt word_le_nat_alt field_simps)
  apply (case_tac "1 + n = 0")
   apply simp_all
  apply (subst(asm) unatSuc, assumption)
  apply arith
  done

lemma plus_one_helper2:
  "⟦ x ≤ n; n + 1 ≠ 0 ⟧ ⟹ x < n + (1 :: 'a :: len word)"
  by (simp add: word_less_nat_alt word_le_nat_alt field_simps
                unatSuc)

lemma less_x_plus_1:
  fixes x :: "'a :: len word" shows
  "x ≠ max_word ⟹ (y < (x + 1)) = (y < x ∨ y = x)"
  apply (rule iffI)
   apply (rule disjCI)
   apply (drule plus_one_helper)
   apply simp
  apply (subgoal_tac "x < x + 1")
   apply (erule disjE, simp_all)
  apply (rule plus_one_helper2 [OF order_refl])
  apply (rule notI, drule max_word_wrap)
  apply simp
  done

lemma word_Suc_leq:
  fixes k::"'a::len word" shows "k ≠ max_word ⟹ x < k + 1 ⟷ x ≤ k"
  using less_x_plus_1 word_le_less_eq by auto

lemma word_Suc_le:
   fixes k::"'a::len word" shows "x ≠ max_word ⟹ x + 1 ≤ k ⟷ x < k"
  by (meson not_less word_Suc_leq)

lemma word_lessThan_Suc_atMost:
  ‹{..< k + 1} = {..k}› if ‹k ≠ - 1› for k :: ‹'a::len word›
  using that by (simp add: lessThan_def atMost_def word_Suc_leq)

lemma word_atLeastLessThan_Suc_atLeastAtMost:
  ‹{l ..< u + 1} = {l..u}› if ‹u ≠ - 1› for l :: ‹'a::len word›
  using that by (simp add: atLeastAtMost_def atLeastLessThan_def word_lessThan_Suc_atMost)

lemma word_atLeastAtMost_Suc_greaterThanAtMost:
  ‹{m<..u} = {m + 1..u}› if ‹m ≠ - 1› for m :: ‹'a::len word›
  using that by (simp add: greaterThanAtMost_def greaterThan_def atLeastAtMost_def atLeast_def word_Suc_le)

lemma word_atLeastLessThan_Suc_atLeastAtMost_union:
  fixes l::"'a::len word"
  assumes "m ≠ max_word" and "l ≤ m" and "m ≤ u"
  shows "{l..m} ∪ {m+1..u} = {l..u}"
  proof -
  from ivl_disj_un_two(8)[OF assms(2) assms(3)] have "{l..u} = {l..m} ∪ {m<..u}" by blast
  with assms show ?thesis by(simp add: word_atLeastAtMost_Suc_greaterThanAtMost)
  qed

lemma max_word_less_eq_iff [simp]:
  ‹- 1 ≤ w ⟷ w = - 1› for w :: ‹'a::len word›
  by (fact word_order.extremum_unique)

lemma word_or_zero:
  "(a OR b = 0) = (a = 0 ∧ b = 0)"
  for a b :: ‹'a::len word›
  by (fact or_eq_0_iff)

lemma word_2p_mult_inc:
  assumes x: "2 * 2 ^ n < (2::'a::len word) * 2 ^ m"
  assumes suc_n: "Suc n < LENGTH('a::len)"
  shows "2^n < (2::'a::len word)^m"
  by (smt suc_n le_less_trans lessI nat_less_le nat_mult_less_cancel_disj p2_gt_0
          power_Suc power_Suc unat_power_lower word_less_nat_alt x)

lemma power_overflow:
  "n ≥ LENGTH('a) ⟹ 2 ^ n = (0 :: 'a::len word)"
  by simp

lemmas extra_sle_sless_unfolds [simp] =
    word_sle_eq[where a=0 and b=1]
    word_sle_eq[where a=0 and b="numeral n"]
    word_sle_eq[where a=1 and b=0]
    word_sle_eq[where a=1 and b="numeral n"]
    word_sle_eq[where a="numeral n" and b=0]
    word_sle_eq[where a="numeral n" and b=1]
    word_sless_alt[where a=0 and b=1]
    word_sless_alt[where a=0 and b="numeral n"]
    word_sless_alt[where a=1 and b=0]
    word_sless_alt[where a=1 and b="numeral n"]
    word_sless_alt[where a="numeral n" and b=0]
    word_sless_alt[where a="numeral n" and b=1]
  for n

lemma word_sint_1:
  "sint (1::'a::len word) = (if LENGTH('a) = 1 then -1 else 1)"
  by (fact signed_1)

lemma ucast_of_nat:
  "is_down (ucast :: 'a :: len word ⇒ 'b :: len word)
    ⟹ ucast (of_nat n :: 'a word) = (of_nat n :: 'b word)"
  by transfer simp

lemma scast_1':
  "(scast (1::'a::len word) :: 'b::len word) =
   (word_of_int (signed_take_bit (LENGTH('a::len) - Suc 0) (1::int)))"
  by transfer simp

lemma scast_1:
  "(scast (1::'a::len word) :: 'b::len word) = (if LENGTH('a) = 1 then -1 else 1)"
  by (fact signed_1)

lemma unat_minus_one_word:
  "unat (-1 :: 'a :: len word) = 2 ^ LENGTH('a) - 1"
  apply (simp only: flip: mask_eq_exp_minus_1)
  apply transfer
  apply (simp add: take_bit_minus_one_eq_mask nat_mask_eq)
  done

lemmas word_diff_ls'' = word_diff_ls [where xa=x and x=x for x]
lemmas word_diff_ls' = word_diff_ls'' [simplified]

lemmas word_l_diffs' = word_l_diffs [where xa=x and x=x for x]
lemmas word_l_diffs = word_l_diffs' [simplified]

lemma two_power_increasing:
  "⟦ n ≤ m; m < LENGTH('a) ⟧ ⟹ (2 :: 'a :: len word) ^ n ≤ 2 ^ m"
  by (simp add: word_le_nat_alt)

lemma word_leq_le_minus_one:
  "⟦ x ≤ y; x ≠ 0 ⟧ ⟹ x - 1 < (y :: 'a :: len word)"
  apply (simp add: word_less_nat_alt word_le_nat_alt)
  apply (subst unat_minus_one)
   apply assumption
  apply (cases "unat x")
   apply (simp add: unat_eq_zero)
  apply arith
  done

lemma neg_mask_combine:
  "NOT(mask a) AND NOT(mask b) = NOT(mask (max a b) :: 'a::len word)"
  by (rule bit_word_eqI) (auto simp add: bit_simps)

lemma neg_mask_twice:
  "x AND NOT(mask n) AND NOT(mask m) = x AND NOT(mask (max n m))"
  for x :: ‹'a::len word›
  by (rule bit_word_eqI) (auto simp add: bit_simps)

lemma multiple_mask_trivia:
  "n ≥ m ⟹ (x AND NOT(mask n)) + (x AND mask n AND NOT(mask m)) = x AND NOT(mask m)"
  for x :: ‹'a::len word›
  apply (rule trans[rotated], rule_tac w="mask n" in word_plus_and_or_coroll2)
  apply (simp add: word_bw_assocs word_bw_comms word_bw_lcs neg_mask_twice
                   max_absorb2)
  done

lemma word_of_nat_less:
  "⟦ n < unat x ⟧ ⟹ of_nat n < x"
  apply (simp add: word_less_nat_alt)
  apply (erule order_le_less_trans[rotated])
  apply (simp add: take_bit_eq_mod)
  done

lemma unat_mask:
  "unat (mask n :: 'a :: len word) = 2 ^ (min n (LENGTH('a))) - 1"
  apply (subst min.commute)
  apply (simp add: mask_eq_decr_exp not_less min_def  split: if_split_asm)
  apply (intro conjI impI)
   apply (simp add: unat_sub_if_size)
   apply (simp add: power_overflow word_size)
  apply (simp add: unat_sub_if_size)
  done

lemma mask_over_length:
  "LENGTH('a) ≤ n ⟹ mask n = (-1::'a::len word)"
  by (simp add: mask_eq_decr_exp)

lemma Suc_2p_unat_mask:
  "n < LENGTH('a) ⟹ Suc (2 ^ n * k + unat (mask n :: 'a::len word)) = 2 ^ n * (k+1)"
  by (simp add: unat_mask)

lemma sint_of_nat_ge_zero:
  "x < 2 ^ (LENGTH('a) - 1) ⟹ sint (of_nat x :: 'a :: len word) ≥ 0"
  by (simp add: bit_iff_odd)

lemma int_eq_sint:
  "x < 2 ^ (LENGTH('a) - 1) ⟹ sint (of_nat x :: 'a :: len word) = int x"
  apply transfer
  apply (rule signed_take_bit_int_eq_self)
   apply simp_all
  apply (metis negative_zle numeral_power_eq_of_nat_cancel_iff)
  done

lemma sint_of_nat_le:
  "⟦ b < 2 ^ (LENGTH('a) - 1); a ≤ b ⟧
   ⟹ sint (of_nat a :: 'a :: len word) ≤ sint (of_nat b :: 'a :: len word)"
  apply (cases ‹LENGTH('a)›)
  apply simp_all
  apply transfer
  apply (subst signed_take_bit_eq_if_positive)
   apply (simp add: bit_simps)
  apply (metis bit_take_bit_iff nat_less_le order_less_le_trans take_bit_nat_eq_self_iff)
  apply (subst signed_take_bit_eq_if_positive)
    apply (simp add: bit_simps)
  apply (metis bit_take_bit_iff nat_less_le take_bit_nat_eq_self_iff)
    apply (simp flip: of_nat_take_bit add: take_bit_nat_eq_self)
  done

lemma word_le_not_less:
  "((b::'a::len word) ≤ a) = (¬(a < b))"
  by fastforce

lemma less_is_non_zero_p1:
  fixes a :: "'a :: len word"
  shows "a < k ⟹ a + 1 ≠ 0"
  apply (erule contrapos_pn)
  apply (drule max_word_wrap)
  apply (simp add: not_less)
  done

lemma unat_add_lem':
  "(unat x + unat y < 2 ^ LENGTH('a)) ⟹
    (unat (x + y :: 'a :: len word) = unat x + unat y)"
  by (subst unat_add_lem[symmetric], assumption)

lemma word_less_two_pow_divI:
  "⟦ (x :: 'a::len word) < 2 ^ (n - m); m ≤ n; n < LENGTH('a) ⟧ ⟹ x < 2 ^ n div 2 ^ m"
  apply (simp add: word_less_nat_alt)
  apply (subst unat_word_ariths)
  apply (subst mod_less)
   apply (rule order_le_less_trans [OF div_le_dividend])
   apply (rule unat_lt2p)
  apply (simp add: power_sub)
  done

lemma word_less_two_pow_divD:
  "⟦ (x :: 'a::len word) < 2 ^ n div 2 ^ m ⟧
     ⟹ n ≥ m ∧ (x < 2 ^ (n - m))"
  apply (cases "n < LENGTH('a)")
   apply (cases "m < LENGTH('a)")
    apply (simp add: word_less_nat_alt)
    apply (subst(asm) unat_word_ariths)
    apply (subst(asm) mod_less)
     apply (rule order_le_less_trans [OF div_le_dividend])
     apply (rule unat_lt2p)
    apply (clarsimp dest!: less_two_pow_divD)
   apply (simp add: power_overflow)
   apply (simp add: word_div_def)
  apply (simp add: power_overflow word_div_def)
  done

lemma of_nat_less_two_pow_div_set:
  "⟦ n < LENGTH('a) ⟧ ⟹
   {x. x < (2 ^ n div 2 ^ m :: 'a::len word)}
      = of_nat ` {k. k < 2 ^ n div 2 ^ m}"
  apply (simp add: image_def)
  apply (safe dest!: word_less_two_pow_divD less_two_pow_divD
             intro!: word_less_two_pow_divI)
   apply (rule_tac x="unat x" in exI)
   apply (simp add: power_sub[symmetric])
   apply (subst unat_power_lower[symmetric, where 'a='a])
    apply simp
   apply (erule unat_mono)
  apply (subst word_unat_power)
  apply (rule of_nat_mono_maybe)
   apply (rule power_strict_increasing)
    apply simp
   apply simp
  apply assumption
  done

lemma ucast_less:
  "LENGTH('b) < LENGTH('a) ⟹
   (ucast (x :: 'b :: len word) :: ('a :: len word)) < 2 ^ LENGTH('b)"
  by transfer simp

lemma ucast_range_less:
  "LENGTH('a :: len) < LENGTH('b :: len) ⟹
   range (ucast :: 'a word ⇒ 'b word) = {x. x < 2 ^ len_of TYPE ('a)}"
  apply safe
   apply (erule ucast_less)
  apply (simp add: image_def)
  apply (rule_tac x="ucast x" in exI)
  apply (rule bit_word_eqI)
  apply (auto simp add: bit_simps)
  apply (metis bit_take_bit_iff less_mask_eq not_less take_bit_eq_mask)
  done

lemma word_power_less_diff:
  "⟦2 ^ n * q < (2::'a::len word) ^ m; q < 2 ^ (LENGTH('a) - n)⟧ ⟹ q < 2 ^ (m - n)"
  apply (case_tac "m ≥ LENGTH('a)")
   apply (simp add: power_overflow)
  apply (case_tac "n ≥ LENGTH('a)")
   apply (simp add: power_overflow)
  apply (cases "n = 0")
   apply simp
  apply (subst word_less_nat_alt)
  apply (subst unat_power_lower)
   apply simp
  apply (rule nat_power_less_diff)
  apply (simp add: word_less_nat_alt)
  apply (subst (asm) iffD1 [OF unat_mult_lem])
   apply (simp add:nat_less_power_trans)
  apply simp
  done

lemma word_less_sub_1:
  "x < (y :: 'a :: len word) ⟹ x ≤ y - 1"
  by (fact word_le_minus_one_leq)

lemma word_sub_mono2:
  "⟦ a + b ≤ c + d; c ≤ a; b ≤ a + b; d ≤ c + d ⟧
    ⟹ b ≤ (d :: 'a :: len word)"
  apply (drule(1) word_sub_mono)
    apply simp
   apply simp
  apply simp
  done

lemma word_not_le:
  "(¬ x ≤ (y :: 'a :: len word)) = (y < x)"
  by fastforce

lemma word_subset_less:
  "⟦ {x .. x + r - 1} ⊆ {y .. y + s - 1};
     x ≤ x + r - 1; y ≤ y + (s :: 'a :: len word) - 1;
     s ≠ 0 ⟧
     ⟹ r ≤ s"
  apply (frule subsetD[where c=x])
   apply simp
  apply (drule subsetD[where c="x + r - 1"])
   apply simp
  apply (clarsimp simp: add_diff_eq[symmetric])
  apply (drule(1) word_sub_mono2)
    apply (simp_all add: olen_add_eqv[symmetric])
  apply (erule word_le_minus_cancel)
  apply (rule ccontr)
  apply (simp add: word_not_le)
  done

lemma uint_power_lower:
  "n < LENGTH('a) ⟹ uint (2 ^ n :: 'a :: len word) = (2 ^ n :: int)"
  by (rule uint_2p_alt)

lemma power_le_mono:
  "⟦2 ^ n ≤ (2::'a::len word) ^ m; n < LENGTH('a); m < LENGTH('a)⟧
   ⟹ n ≤ m"
  apply (clarsimp simp add: le_less)
  apply safe
  apply (simp add: word_less_nat_alt)
  apply (simp only: uint_arith_simps(3))
  apply (drule uint_power_lower)+
  apply simp
  done

lemma two_power_eq:
  "⟦n < LENGTH('a); m < LENGTH('a)⟧
   ⟹ ((2::'a::len word) ^ n = 2 ^ m) = (n = m)"
  apply safe
  apply (rule order_antisym)
   apply (simp add: power_le_mono[where 'a='a])+
  done

lemma unat_less_helper:
  "x < of_nat n ⟹ unat x < n"
  apply (simp add: word_less_nat_alt)
  apply (erule order_less_le_trans)
  apply (simp add: take_bit_eq_mod)
  done

lemma nat_uint_less_helper:
  "nat (uint y) = z ⟹ x < y ⟹ nat (uint x) < z"
  apply (erule subst)
  apply (subst unat_eq_nat_uint [symmetric])
  apply (subst unat_eq_nat_uint [symmetric])
  by (simp add: unat_mono)

lemma of_nat_0:
  "⟦of_nat n = (0::'a::len word); n < 2 ^ LENGTH('a)⟧ ⟹ n = 0"
  by transfer (simp add: take_bit_eq_mod)

lemma of_nat_inj:
  "⟦x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)⟧ ⟹
   (of_nat x = (of_nat y :: 'a :: len word)) = (x = y)"
  by (metis unat_of_nat_len)

lemma div_to_mult_word_lt:
  "⟦ (x :: 'a :: len word) ≤ y div z ⟧ ⟹ x * z ≤ y"
  apply (cases "z = 0")
   apply simp
  apply (simp add: word_neq_0_conv)
  apply (rule order_trans)
   apply (erule(1) word_mult_le_mono1)
   apply (simp add: unat_div)
   apply (rule order_le_less_trans [OF div_mult_le])
   apply simp
  apply (rule word_div_mult_le)
  done

lemma ucast_ucast_mask:
  "(ucast :: 'a :: len word ⇒ 'b :: len word) (ucast x) = x AND mask (len_of TYPE ('a))"
  apply (simp flip: take_bit_eq_mask)
  apply transfer
  apply (simp add: ac_simps)
  done

lemma ucast_ucast_len:
  "⟦ x < 2 ^ LENGTH('b) ⟧ ⟹ ucast (ucast x::'b::len word) = (x::'a::len word)"
  apply (subst ucast_ucast_mask)
  apply (erule less_mask_eq)
  done

lemma ucast_ucast_id:
  "LENGTH('a) < LENGTH('b) ⟹ ucast (ucast (x::'a::len word)::'b::len word) = x"
  by (auto intro: ucast_up_ucast_id simp: is_up_def source_size_def target_size_def word_size)

lemma unat_ucast:
  "unat (ucast x :: ('a :: len) word) = unat x mod 2 ^ (LENGTH('a))"
proof -
  have ‹2 ^ LENGTH('a) = nat (2 ^ LENGTH('a))›
    by simp
  moreover have ‹unat (ucast x :: 'a word) = unat x mod nat (2 ^ LENGTH('a))›
    by transfer (simp flip: nat_mod_distrib take_bit_eq_mod)
  ultimately show ?thesis
    by (simp only:)
qed

lemma ucast_less_ucast:
  "LENGTH('a) ≤ LENGTH('b) ⟹
   (ucast x < ((ucast (y :: 'a::len word)) :: 'b::len word)) = (x < y)"
  apply (simp add: word_less_nat_alt unat_ucast)
  apply (subst mod_less)
   apply(rule less_le_trans[OF unat_lt2p], simp)
  apply (subst mod_less)
   apply(rule less_le_trans[OF unat_lt2p], simp)
  apply simp
  done

― ‹This weaker version was previously called @{text ucast_less_ucast}. We retain it to
    support existing proofs.›
lemmas ucast_less_ucast_weak = ucast_less_ucast[OF order.strict_implies_order]

lemma unat_Suc2:
  fixes n :: "'a :: len word"
  shows
  "n ≠ -1 ⟹ unat (n + 1) = Suc (unat n)"
  apply (subst add.commute, rule unatSuc)
  apply (subst eq_diff_eq[symmetric], simp add: minus_equation_iff)
  done

lemma word_div_1:
  "(n :: 'a :: len word) div 1 = n"
  by (fact bits_div_by_1)

lemma word_minus_one_le:
  "-1 ≤ (x :: 'a :: len word) = (x = -1)"
  by (fact word_order.extremum_unique)

lemma up_scast_inj:
      "⟦ scast x = (scast y :: 'b :: len word); size x ≤ LENGTH('b) ⟧
         ⟹ x = y"
  apply transfer
  apply (cases ‹LENGTH('a)›)
  apply simp_all
  apply (metis order_refl take_bit_signed_take_bit take_bit_tightened)
  done

lemma up_scast_inj_eq:
  "LENGTH('a) ≤ len_of TYPE ('b) ⟹
  (scast x = (scast y::'b::len word)) = (x = (y::'a::len word))"
  by (fastforce dest: up_scast_inj simp: word_size)

lemma word_le_add:
  fixes x :: "'a :: len word"
  shows "x ≤ y ⟹ ∃n. y = x + of_nat n"
  by (rule exI [where x = "unat (y - x)"]) simp

lemma word_plus_mcs_4':
  fixes x :: "'a :: len word"
  shows "⟦x + v ≤ x + w; x ≤ x + v⟧ ⟹ v ≤ w"
  apply (rule word_plus_mcs_4)
   apply (simp add: add.commute)
  apply (simp add: add.commute)
  done

lemma unat_eq_1:
  ‹unat x = Suc 0 ⟷ x = 1›
  by (auto intro!: unsigned_word_eqI [where ?'a = nat])

lemma word_unat_Rep_inject1:
  ‹unat x = unat 1 ⟷ x = 1›
  by (simp add: unat_eq_1)

lemma and_not_mask_twice:
  "(w AND NOT (mask n)) AND NOT (mask m) = w AND NOT (mask (max m n))"
  for w :: ‹'a::len word›
  by (rule bit_word_eqI) (auto simp add: bit_simps)

lemma word_less_cases:
  "x < y ⟹ x = y - 1 ∨ x < y - (1 ::'a::len word)"
  apply (drule word_less_sub_1)
  apply (drule order_le_imp_less_or_eq)
  apply auto
  done

lemma mask_and_mask:
  "mask a AND mask b = (mask (min a b) :: 'a::len word)"
  by (simp flip: take_bit_eq_mask ac_simps)

lemma mask_eq_0_eq_x:
  "(x AND w = 0) = (x AND NOT w = x)"
  for x w :: ‹'a::len word›
  using word_plus_and_or_coroll2[where x=x and w=w]
  by auto

lemma mask_eq_x_eq_0:
  "(x AND w = x) = (x AND NOT w = 0)"
  for x w :: ‹'a::len word›
  using word_plus_and_or_coroll2[where x=x and w=w]
  by auto

lemma compl_of_1: "NOT 1 = (-2 :: 'a :: len word)"
  by (fact not_one)

lemma split_word_eq_on_mask:
  "(x = y) = (x AND m = y AND m ∧ x AND NOT m = y AND NOT m)"
  for x y m :: ‹'a::len word›
  apply transfer
  apply (simp add: bit_eq_iff)
  apply (auto simp add: bit_simps ac_simps)
  done

lemma word_FF_is_mask:
  "0xFF = (mask 8 :: 'a::len word)"
  by (simp add: mask_eq_decr_exp)

lemma word_1FF_is_mask:
  "0x1FF = (mask 9 :: 'a::len word)"
  by (simp add: mask_eq_decr_exp)

lemma ucast_of_nat_small:
  "x < 2 ^ LENGTH('a) ⟹ ucast (of_nat x :: 'a :: len word) = (of_nat x :: 'b :: len word)"
  apply transfer
  apply (auto simp add: take_bit_of_nat min_def not_le)
  apply (metis linorder_not_less min_def take_bit_nat_eq_self take_bit_take_bit)
  done

lemma word_le_make_less:
  fixes x :: "'a :: len word"
  shows "y ≠ -1 ⟹ (x ≤ y) = (x < (y + 1))"
  apply safe
  apply (erule plus_one_helper2)
  apply (simp add: eq_diff_eq[symmetric])
  done

lemmas finite_word = finite [where 'a="'a::len word"]

lemma word_to_1_set:
  "{0 ..< (1 :: 'a :: len word)} = {0}"
  by fastforce

lemma word_leq_minus_one_le:
  fixes x :: "'a::len word"
  shows "⟦y ≠ 0; x ≤ y - 1 ⟧ ⟹ x < y"
  using le_m1_iff_lt word_neq_0_conv by blast

lemma word_count_from_top:
  "n ≠ 0 ⟹ {0 ..< n :: 'a :: len word} = {0 ..< n - 1} ∪ {n - 1}"
  apply (rule set_eqI, rule iffI)
   apply simp
   apply (drule word_le_minus_one_leq)
   apply (rule disjCI)
   apply simp
  apply simp
  apply (erule word_leq_minus_one_le)
  apply fastforce
  done

lemma word_minus_one_le_leq:
  "⟦ x - 1 < y ⟧ ⟹ x ≤ (y :: 'a :: len word)"
  apply (cases "x = 0")
   apply simp
  apply (simp add: word_less_nat_alt word_le_nat_alt)
  apply (subst(asm) unat_minus_one)
   apply (simp add: word_less_nat_alt)
  apply (cases "unat x")
   apply (simp add: unat_eq_zero)
  apply arith
  done

lemma word_div_less:
  "m < n ⟹ m div n = 0" for m :: "'a :: len word"
  by (simp add: unat_mono word_arith_nat_defs(6))

lemma word_must_wrap:
  "⟦ x ≤ n - 1; n ≤ x ⟧ ⟹ n = (0 :: 'a :: len word)"
  using dual_order.trans sub_wrap word_less_1 by blast

lemma range_subset_card:
  "⟦ {a :: 'a :: len word .. b} ⊆ {c .. d}; b ≥ a ⟧ ⟹ d ≥ c ∧ d - c ≥ b - a"
  using word_sub_le word_sub_mono by fastforce

lemma less_1_simp:
  "n - 1 < m = (n ≤ (m :: 'a :: len word) ∧ n ≠ 0)"
  by unat_arith

lemma word_power_mod_div:
  fixes x :: "'a::len word"
  shows "⟦ n < LENGTH('a); m < LENGTH('a)⟧
  ⟹ x mod 2 ^ n div 2 ^ m = x div 2 ^ m mod 2 ^ (n - m)"
  apply (simp add: word_arith_nat_div unat_mod power_mod_div)
  apply (subst unat_arith_simps(3))
  apply (subst unat_mod)
  apply (subst unat_of_nat)+
  apply (simp add: mod_mod_power min.commute)
  done

lemma word_range_minus_1':
  fixes a :: "'a :: len word"
  shows "a ≠ 0 ⟹ {a - 1<..b} = {a..b}"
  by (simp add: greaterThanAtMost_def atLeastAtMost_def greaterThan_def atLeast_def less_1_simp)

lemma word_range_minus_1:
  fixes a :: "'a :: len word"
  shows "b ≠ 0 ⟹ {a..b - 1} = {a..<b}"
  apply (simp add: atLeastLessThan_def atLeastAtMost_def atMost_def lessThan_def)
  apply (rule arg_cong [where f = "λx. {a..} ∩ x"])
  apply rule
   apply clarsimp
   apply (erule contrapos_pp)
   apply (simp add: linorder_not_less linorder_not_le word_must_wrap)
  apply (clarsimp)
  apply (drule word_le_minus_one_leq)
  apply (auto simp: word_less_sub_1)
  done

lemma ucast_nat_def:
  "of_nat (unat x) = (ucast :: 'a :: len word ⇒ 'b :: len word) x"
  by transfer simp

lemma overflow_plus_one_self:
  "(1 + p ≤ p) = (p = (-1 :: 'a :: len word))"
  apply rule
  apply (rule ccontr)
   apply (drule plus_one_helper2)
   apply (rule notI)
   apply (drule arg_cong[where f="λx. x - 1"])
   apply simp
   apply (simp add: field_simps)
  apply simp
  done

lemma plus_1_less:
  "(x + 1 ≤ (x :: 'a :: len word)) = (x = -1)"
  apply (rule iffI)
   apply (rule ccontr)
   apply (cut_tac plus_one_helper2[where x=x, OF order_refl])
    apply simp
   apply clarsimp
   apply (drule arg_cong[where f="λx. x - 1"])
   apply simp
  apply simp
  done

lemma pos_mult_pos_ge:
  "[|x > (0::int); n>=0 |] ==> n * x >= n*1"
  apply (simp only: mult_left_mono)
  done

lemma word_plus_strict_mono_right:
  fixes x :: "'a :: len word"
  shows "⟦y < z; x ≤ x + z⟧ ⟹ x + y < x + z"
  by unat_arith

lemma word_div_mult:
  "0 < c ⟹ a < b * c ⟹ a div c < b" for a b c :: "'a::len word"
  by (rule classical)
     (use div_to_mult_word_lt [of b a c] in
      ‹auto simp add: word_less_nat_alt word_le_nat_alt unat_div›)

lemma word_less_power_trans_ofnat:
  "⟦n < 2 ^ (m - k); k ≤ m; m < LENGTH('a)⟧
   ⟹ of_nat n * 2 ^ k < (2::'a::len word) ^ m"
  apply (subst mult.commute)
  apply (rule word_less_power_trans)
    apply (simp_all add: word_less_nat_alt less_le_trans take_bit_eq_mod)
  done

lemma word_1_le_power:
  "n < LENGTH('a) ⟹ (1 :: 'a :: len word) ≤ 2 ^ n"
  by (rule inc_le[where i=0, simplified], erule iffD2[OF p2_gt_0])

lemma unat_1_0:
  "1 ≤ (x::'a::len word) = (0 < unat x)"
  by (auto simp add: word_le_nat_alt)

lemma x_less_2_0_1':
  fixes x :: "'a::len word"
  shows "⟦LENGTH('a) ≠ 1; x < 2⟧ ⟹ x = 0 ∨ x = 1"
  apply (cases ‹2 ≤ LENGTH('a)›)
  apply simp_all
  apply transfer
  apply auto
  apply (metis add.commute add.right_neutral even_two_times_div_two mod_div_trivial mod_pos_pos_trivial mult.commute mult_zero_left not_less not_take_bit_negative odd_two_times_div_two_succ)
  done

lemmas word_add_le_iff2 = word_add_le_iff [folded no_olen_add_nat]

lemma of_nat_power:
  shows "⟦ p < 2 ^ x; x < len_of TYPE ('a) ⟧ ⟹ of_nat p < (2 :: 'a :: len word) ^ x"
  apply (rule order_less_le_trans)
   apply (rule of_nat_mono_maybe)
    apply (erule power_strict_increasing)
    apply simp
   apply assumption
  apply (simp add: word_unat_power del: of_nat_power)
  done

lemma of_nat_n_less_equal_power_2:
  "n < LENGTH('a::len) ⟹ ((of_nat n)::'a word) < 2 ^ n"
  apply (induct n)
   apply clarsimp
  apply clarsimp
  apply (metis of_nat_power n_less_equal_power_2 of_nat_Suc power_Suc)
  done

lemma eq_mask_less:
  fixes w :: "'a::len word"
  assumes eqm: "w = w AND mask n"
  and      sz: "n < len_of TYPE ('a)"
  shows "w < (2::'a word) ^ n"
  by (subst eqm, rule and_mask_less' [OF sz])

lemma of_nat_mono_maybe':
  fixes Y :: "nat"
  assumes xlt: "x < 2 ^ len_of TYPE ('a)"
  assumes ylt: "y < 2 ^ len_of TYPE ('a)"
  shows   "(y < x) = (of_nat y < (of_nat x :: 'a :: len word))"
  apply (subst word_less_nat_alt)
  apply (subst unat_of_nat)+
  apply (subst mod_less)
   apply (rule ylt)
  apply (subst mod_less)
   apply (rule xlt)
  apply simp
  done

lemma of_nat_mono_maybe_le:
  "⟦x < 2 ^ LENGTH('a); y < 2 ^ LENGTH('a)⟧ ⟹
  (y ≤ x) = ((of_nat y :: 'a :: len word) ≤ of_nat x)"
  apply (clarsimp simp: le_less)
  apply (rule disj_cong)
   apply (rule of_nat_mono_maybe', assumption+)
  apply auto
  using of_nat_inj apply blast
  done

lemma mask_AND_NOT_mask:
  "(w AND NOT (mask n)) AND mask n = 0"
  for w :: ‹'a::len word›
  by (rule bit_word_eqI) (simp add: bit_simps)

lemma AND_NOT_mask_plus_AND_mask_eq:
  "(w AND NOT (mask n)) + (w AND mask n) = w"
  for w :: ‹'a::len word›
  apply (subst disjunctive_add)
  apply (auto simp add: bit_simps)
  apply (rule bit_word_eqI)
  apply (auto simp add: bit_simps)
  done

lemma mask_eqI:
  fixes x :: "'a :: len word"
  assumes m1: "x AND mask n = y AND mask n"
  and     m2: "x AND NOT (mask n) = y AND NOT (mask n)"
  shows "x = y"
proof -
  have *: ‹x = x AND mask n OR x AND NOT (mask n)› for x :: ‹'a word›
    by (rule bit_word_eqI) (auto simp add: bit_simps)
  from assms * [of x] * [of y] show ?thesis
    by simp
qed

lemma neq_0_no_wrap:
  fixes x :: "'a :: len word"
  shows "⟦ x ≤ x + y; x ≠ 0 ⟧ ⟹ x + y ≠ 0"
  by clarsimp

lemma unatSuc2:
  fixes n :: "'a :: len word"
  shows "n + 1 ≠ 0 ⟹ unat (n + 1) = Suc (unat n)"
  by (simp add: add.commute unatSuc)

lemma word_of_nat_le:
  "n ≤ unat x ⟹ of_nat n ≤ x"
  apply (simp add: word_le_nat_alt unat_of_nat)
  apply (erule order_trans[rotated])
  apply (simp add: take_bit_eq_mod)
  done

lemma word_unat_less_le:
  "a ≤ of_nat b ⟹ unat a ≤ b"
  by (metis eq_iff le_cases le_unat_uoi word_of_nat_le)

lemma mask_Suc_0 : "mask (Suc 0) = (1 :: 'a::len word)"
  by (simp add: mask_eq_decr_exp)

lemma bool_mask':
  fixes x :: "'a :: len word"
  shows "2 < LENGTH('a) ⟹ (0 < x AND 1) = (x AND 1 = 1)"
  by (simp add: and_one_eq mod_2_eq_odd)

lemma ucast_ucast_add:
  fixes x :: "'a :: len word"
  fixes y :: "'b :: len word"
  shows
  "LENGTH('b) ≥ LENGTH('a) ⟹
    ucast (ucast x + y) = x + ucast y"
  apply transfer
  apply simp
  apply (subst (2) take_bit_add [symmetric])
  apply (subst take_bit_add [symmetric])
  apply simp
  done

lemma lt1_neq0:
  fixes x :: "'a :: len word"
  shows "(1 ≤ x) = (x ≠ 0)" by unat_arith

lemma word_plus_one_nonzero:
  fixes x :: "'a :: len word"
  shows "⟦x ≤ x + y; y ≠ 0⟧ ⟹ x + 1 ≠ 0"
  apply (subst lt1_neq0 [symmetric])
  apply (subst olen_add_eqv [symmetric])
  apply (erule word_random)
  apply (simp add: lt1_neq0)
  done

lemma word_sub_plus_one_nonzero:
  fixes n :: "'a :: len word"
  shows "⟦n' ≤ n; n' ≠ 0⟧ ⟹ (n - n') + 1 ≠ 0"
  apply (subst lt1_neq0 [symmetric])
  apply (subst olen_add_eqv [symmetric])
  apply (rule word_random [where x' = n'])
   apply simp
   apply (erule word_sub_le)
  apply (simp add: lt1_neq0)
  done

lemma word_le_minus_mono_right:
  fixes x :: "'a :: len word"
  shows "⟦ z ≤ y; y ≤ x; z ≤ x ⟧ ⟹ x - y ≤ x - z"
  apply (rule word_sub_mono)
     apply simp
    apply assumption
   apply (erule word_sub_le)
  apply (erule word_sub_le)
  done

lemma word_0_sle_from_less:
  ‹0 ≤s x› if ‹x < 2 ^ (LENGTH('a) - 1)› for x :: ‹'a::len word›
  using that
  apply transfer
  apply (cases ‹LENGTH('a)›)
   apply simp_all
  apply (metis bit_take_bit_iff min_def nat_less_le not_less_eq take_bit_int_eq_self_iff take_bit_take_bit)
  done

lemma ucast_sub_ucast:
  fixes x :: "'a::len word"
  assumes "y ≤ x"
  assumes T: "LENGTH('a) ≤ LENGTH('b)"
  shows "ucast (x - y) = (ucast x - ucast y :: 'b::len word)"
proof -
  from T
  have P: "unat x < 2 ^ LENGTH('b)" "unat y < 2 ^ LENGTH('b)"
    by (fastforce intro!: less_le_trans[OF unat_lt2p])+
  then show ?thesis
    by (simp add: unat_arith_simps unat_ucast assms[simplified unat_arith_simps])
qed

lemma word_1_0:
  "⟦a + (1::('a::len) word) ≤ b; a < of_nat x⟧ ⟹ a < b"
  apply transfer
  apply (subst (asm) take_bit_incr_eq)
   apply (auto simp add: diff_less_eq)
  using take_bit_int_less_exp le_less_trans by blast

lemma unat_of_nat_less:"⟦ a < b; unat b = c ⟧ ⟹ a < of_nat c"
  by fastforce

lemma word_le_plus_1: "⟦ (y::('a::len) word) < y + n; a < n ⟧ ⟹ y + a ≤ y + a + 1"
  by unat_arith

lemma word_le_plus:"⟦(a::('a::len) word) < a + b; c < b⟧ ⟹ a ≤ a + c"
  by (metis order_less_imp_le word_random)

lemma sint_minus1 [simp]: "(sint x = -1) = (x = -1)"
  apply (cases ‹LENGTH('a)›)
   apply simp_all
  apply transfer
  apply (simp flip: signed_take_bit_eq_iff_take_bit_eq)
  done

lemma sint_0 [simp]: "(sint x = 0) = (x = 0)"
  by (fact signed_eq_0_iff)

(* It is not always that case that "sint 1 = 1", because of 1-bit word sizes.
 * This lemma produces the different cases. *)
lemma sint_1_cases:
  P if ‹⟦ len_of TYPE ('a::len) = 1; (a::'a word) = 0; sint a = 0 ⟧ ⟹ P›
     ‹⟦ len_of TYPE ('a) = 1; a = 1; sint (1 :: 'a word) = -1 ⟧ ⟹ P›
     ‹⟦ len_of TYPE ('a) > 1; sint (1 :: 'a word) = 1 ⟧ ⟹ P›
proof (cases ‹LENGTH('a) = 1›)
  case True
  then have ‹a = 0 ∨ a = 1›
    by transfer auto
  with True that show ?thesis
    by auto
next
  case False
  with that show ?thesis
    by (simp add: less_le Suc_le_eq)
qed

lemma sint_int_min:
  "sint (- (2 ^ (LENGTH('a) - Suc 0)) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))"
  apply (cases ‹LENGTH('a)›)
   apply simp_all
  apply transfer
  apply (simp add: signed_take_bit_int_eq_self)
  done

lemma sint_int_max_plus_1:
  "sint (2 ^ (LENGTH('a) - Suc 0) :: ('a::len) word) = - (2 ^ (LENGTH('a) - Suc 0))"
  apply (cases ‹LENGTH('a)›)
   apply simp_all
  apply (subst word_of_int_2p [symmetric])
  apply (subst int_word_sint)
  apply simp
  done

lemma uint_range':
  ‹0 ≤ uint x ∧ uint x < 2 ^ LENGTH('a)›
  for x :: ‹'a::len word›
  by transfer simp

lemma sint_of_int_eq:
  "⟦ - (2 ^ (LENGTH('a) - 1)) ≤ x; x < 2 ^ (LENGTH('a) - 1) ⟧ ⟹ sint (of_int x :: ('a::len) word) = x"
  by (simp add: signed_take_bit_int_eq_self)

lemma of_int_sint:
  "of_int (sint a) = a"
  by simp

lemma sint_ucast_eq_uint:
    "⟦ ¬ is_down (ucast :: ('a::len word ⇒ 'b::len word)) ⟧
            ⟹ sint ((ucast :: ('a::len word ⇒ 'b::len word)) x) = uint x"
  apply transfer
  apply (simp add: signed_take_bit_take_bit)
  done

lemma word_less_nowrapI':
  "(x :: 'a :: len word) ≤ z - k ⟹ k ≤ z ⟹ 0 < k ⟹ x < x + k"
  by uint_arith

lemma mask_plus_1:
  "mask n + 1 = (2 ^ n :: 'a::len word)"
  by (clarsimp simp: mask_eq_decr_exp)

lemma unat_inj: "inj unat"
  by (metis eq_iff injI word_le_nat_alt)

lemma unat_ucast_upcast:
  "is_up (ucast :: 'b word ⇒ 'a word)
      ⟹ unat (ucast x :: ('a::len) word) = unat (x :: ('b::len) word)"
  unfolding ucast_eq unat_eq_nat_uint
  apply transfer
  apply simp
  done

lemma ucast_mono:
  "⟦ (x :: 'b :: len word) < y; y < 2 ^ LENGTH('a) ⟧
   ⟹ ucast x < ((ucast y) :: 'a :: len word)"
  apply (simp only: flip: ucast_nat_def)
  apply (rule of_nat_mono_maybe)
  apply (rule unat_less_helper)
  apply simp
  apply (simp add: word_less_nat_alt)
  done

lemma ucast_mono_le:
  "⟦x ≤ y; y < 2 ^ LENGTH('b)⟧ ⟹ (ucast (x :: 'a :: len word) :: 'b :: len word) ≤ ucast y"
  apply (simp only: flip: ucast_nat_def)
  apply (subst of_nat_mono_maybe_le[symmetric])
    apply (rule unat_less_helper)
    apply simp
   apply (rule unat_less_helper)
   apply (erule le_less_trans)
  apply (simp_all add: word_le_nat_alt)
  done

lemma ucast_mono_le':
  "⟦ unat y < 2 ^ LENGTH('b); LENGTH('b::len) < LENGTH('a::len); x ≤ y ⟧
   ⟹ ucast x ≤ (ucast y :: 'b word)" for x y :: ‹'a::len word›
  by (auto simp: word_less_nat_alt intro: ucast_mono_le)

lemma neg_mask_add_mask:
  "((x:: 'a :: len word) AND NOT (mask n)) + (2 ^ n - 1) = x OR mask n"
  unfolding mask_2pm1 [symmetric]
  apply (subst word_plus_and_or_coroll; rule bit_word_eqI)
   apply (auto simp add: bit_simps)
  done

lemma le_step_down_word:"⟦(i::('a::len) word) ≤ n; i = n ⟶ P; i ≤ n - 1 ⟶ P⟧ ⟹ P"
  by unat_arith

lemma le_step_down_word_2:
  fixes x :: "'a::len word"
  shows "⟦x ≤  y; x ≠ y⟧ ⟹ x ≤ y - 1"
  by (subst (asm) word_le_less_eq,
      clarsimp,
      simp add: word_le_minus_one_leq)

lemma NOT_mask_AND_mask[simp]: "(w AND mask n) AND NOT (mask n) = 0"
  by (clarsimp simp add: mask_eq_decr_exp Parity.bit_eq_iff bit_and_iff bit_not_iff bit_mask_iff)

lemma and_and_not[simp]:"(a AND b) AND NOT b = 0"
  for a b :: ‹'a::len word›
  apply (subst word_bw_assocs(1))
  apply clarsimp
  done

lemma ex_mask_1[simp]: "(∃x. mask x = (1 :: 'a::len word))"
  apply (rule_tac x=1 in exI)
  apply (simp add:mask_eq_decr_exp)
  done

lemma not_switch:"NOT a = x ⟹ a = NOT x"
  by auto

end